Systems and methods for MIMO precoding in an xDSL system

ABSTRACT

One embodiment is a method for precoding data for transmission in a discrete multi-tone (DMT) system to cancel self-induced far end crosstalk (self-FEXT). The method comprises learning, by the system, characteristics associated with a plurality of N users within a digital subscriber line (xDSL) system to determine an initial off-diagonal multiple input multiple output (MIMO) precoder (ODMP) for a given tone frequency and converging towards an ODMP from the initial ODMP to cancel self-FEXT for the plurality of N users, wherein the ODMP is represented as a zero diagonal matrix containing only off-diagonal terms.

CROSS-REFERENCE TO RELATED APPLICATION

This application is a continuation of U.S. utility application entitled“Systems and Methods for MIMO Precoding in an xDSL System,” having Ser.No. 11/845,040, filed on Aug. 25, 2007, issued as U.S. Pat. No.7,957,477 on Jun. 7, 2011, which claims priority to, and the benefit of,U.S. Provisional Patent Application entitled, “Small variations MIMOPrecoders SVMP for Downstream FEXT Mitigation,” having Ser. No.60/823,633, filed on Aug. 25, 2006; U.S. Provisional Patent Applicationentitled “Small Variations MIMO Precoders SVMP for Downstream FEXTMitigation,” having Ser. No. 60/854,742, filed Oct. 27, 2006; U.S.Provisional Patent Application entitled “Off Diagonal MIMO Precoder ODMPfor Cooperative Self Fext Cancellation,” having Ser. No. 60/902,764,filed on Feb. 22, 2007; and U.S. Provisional Patent Application entitled“Adaptive Off-Diagonal MIMO Pre-Coder (ODMP) for Downstream DSL SelfFEXT Cancellation,” having Ser. No. 60/894,847, filed Mar. 14, 2007,which are all incorporated by reference herein in their entireties.

TECHNICAL FIELD

The present disclosure generally relates to communications systems andmore particularly, relates to multiple input multiple output (MIMO)precoding in an xDSL system.

BACKGROUND

Industries related to modern communication systems have experienced atremendous growth due to the increasing popularity of the Internet.Digital subscriber line (xDSL) technology is a technology that hasdeveloped in recent years in response to the demand for high-speedInternet access. xDSL technology utilizes the communication medium ofpre-existing telephone systems. Thus, both plain old telephone systems(POTS) and xDSL systems share a common line for xDSL-compatible customerpremises. Similarly, other services such as time compressionmultiplexing (TCM) integrated services digital network (ISDN) can alsoshare a common line with xDSL and POTS.

POTS services and xDSL services are deployed on non-overlappingfrequency bands available on the communication medium. While there isgenerally little concern of cross-talk or other interference betweenPOTS services and xDSL services, xDSL and TCM-ISDN often share a portionof the available bandwidth, thereby making xDSL services susceptible tocross-talk from TCM-ISDN services, and vice versa.

In the telecommunication art, the term “crosstalk” refers tointerference that enters a message channel from one or more otherchannels through a path coupling the message channel with theinterfering channels. Crosstalk can create annoyance in a voice systemor errors in a data system. The degree in which crosstalk impacts thecommunication line will depend in part on such factors as the listener'shearing acuity, extraneous noise on the communication line, thefrequency response of the coupling path, and the level of the disturbingsignal.

There are generally two types of crosstalk mechanisms that arecharacterized, one being far end crosstalk (FEXT) and the other onebeing near-end crosstalk (NEXT). FEXT refers to electromagnetic couplingthat occurs when the receiver on a disturbed pair is located at the farend of the communication line as the transmitter of a disturbing pair.Self-FEXT generally refers to FEXT generated from use of the same timeslot or frequency. In contrast, NEXT results from a disturbing sourceconnected at one end of the wire pair which causes interference in themessage channel at the same end as the disturbing source.

Allocations of wire pairs within telephone cables in accordance withservice requests have typically resulted in a random distribution ofpair utilization with few precise records of actual configurations.Because of the physical proximity of bundled cables (due to pairtwisting, cable branching, cable splicing, etc.), crosstalk caused bythe electromagnetic interference between the neighboring lines is oftenthe dominating noise source in the transmission environment. Inaddition, due to pair twisting in cables where cable branching andsplicing take place, a wire pair can be in close proximity to manydifferent pairs spanning different portions of its length. At atelephone CO (central office), pairs in close proximity may carrydiverse types of service using various modulation schemes, withconsiderable differences in signal levels (and receiver sensitivities)especially for pairs of considerably different lengths.

Both FEXT and self-FEXT (as well as NEXT) continue to be a problem inxDSL communication systems as crosstalk impacts overall performance.Current approaches to addressing crosstalk suffer from various perceivedshortcomings such as increased design costs and inefficient use ofcomputing resources. Therefore, a heretofore unaddressed need exists inthe industry to address the aforementioned deficiencies andinadequacies.

SUMMARY

Briefly described, one embodiment is a method for precoding data fortransmission in a discrete multi-tone (DMT) system to cancelself-induced far end crosstalk (self-FEXT). The method compriseslearning, by the system, characteristics associated with a plurality ofN users within a digital subscriber line (xDSL) system to determine aninitial off-diagonal multiple input multiple output (MIMO) precoder(ODMP) for a given tone frequency and converging towards an ODMP fromthe initial ODMP to cancel self-FEXT for the plurality of N users,wherein the ODMP is represented as a zero diagonal matrix containingonly off-diagonal terms.

Another embodiment is an off-diagonal multiple input multiple output(MIMO) precoder (ODMP) for generating precoded signals to increasesystem performance for a plurality of users in an digital subscriberline (xDSL) system. The ODMP comprises an initialization moduleconfigured to learn characteristics of channels located within the xDSLsystem associated with the plurality of users to derive an initial ODMPand a tracking module configured to converge towards an ODMP from theinitial ODMP to reduce downstream self-induced far end crosstalk(self-FEXT) by executing a least means square (LMS) adaptive algorithm,wherein the ODMP is represented as a zero diagonal matrix containingonly off-diagonal terms.

Another embodiment is an off-diagonal MIMO precoder (ODMP) system in asystem for pre-compensating for downstream self-induced crosstalk(self-FEXT). The system comprises a learning module configured to learnCPE channel characteristics to obtain an initial ODMP, a tracking moduleconfigured to update the initial ODMP during showtime to obtain an ODMPby receiving error data over a feedback channel and executing a leastmeans square (LMS) adaptive algorithm, and an incrementing moduleconfigured to perform one of: add one or more new CPE to the system andremove existing CPE from a digital subscriber line (xDSL) system.

Other systems, methods, features, and advantages of the presentdisclosure will be or become apparent to one with skill in the art uponexamination of the following drawings and detailed description. It isintended that all such additional systems, methods, features, andadvantages be included within this description, be within the scope ofthe present disclosure, and be protected by the accompanying claims.

BRIEF DESCRIPTION OF THE DRAWINGS

Many aspects of the disclosure can be better understood with referenceto the following drawings. The components in the drawings are notnecessarily to scale, emphasis instead being placed upon clearlyillustrating the principles of the present disclosure. Moreover, in thedrawings, like reference numerals designate corresponding partsthroughout the several views.

FIG. 1A illustrates an xDSL system in which embodiments of ODMP areapplied.

FIG. 1B illustrates various components of an alternative embodiment ofthe ODMP shown in FIG. 1A.

FIG. 2 depicts a top-level diagram for an embodiment of a method forderiving an optimum ODMP as described for FIG. 1A.

FIG. 3 illustrates how a low speed back channel may be used toincorporate an embodiment of the method shown in FIG. 2.

DETAILED DESCRIPTION

Having summarized various aspects of the present disclosure, referencewill now be made in detail to the description of the disclosure asillustrated in the drawings. While the disclosure will be described inconnection with these drawings, there is no intent to limit it to theembodiment or embodiments disclosed herein. On the contrary, the intentis to cover all alternatives, modifications and equivalents includedwithin the spirit and scope of the disclosure as defined by the appendedclaims.

By way of background, there are various standards for ADSL systems thatare set forth by the International Telecommunications Union,Telecommunication Standardization Section (ITU-T). One ADSL standard isdescribed in ITU-T Recommendation G.992.1—“Asymmetric Digital SubscriberLine (ADSL) Transceivers”, which is herein incorporated by reference inits entirety. Prior to any transmission of actual data between thecentral office (CO) and the customer premises equipment (CPE), the twoentities must first undergo an initialization procedure designed tofamiliarize the two entities with each other, identify the bandwidthcapabilities for the current session, and further facilitate theestablishment of a valid connection. Pursuant to ADSL standards providedby the International Telecommunication Union-TelecommunicationStandardization Sector (ITU-T), these initialization procedures comprisethe following: 1) a handshake procedure; 2) a transceiver trainingsession; 3) a channel analysis session; 4) an exchange session; andfinally 5) an actual data transmission session commonly referred to as“showtime.”

The present invention relates to DMT based xDSL systems, such as ADSLsystems, very high bit rate xDSL (VDSL) and VDSL2 systems, for example.Various embodiments of the present invention seek to mitigate theeffects of both FEXT and self-FEXT in xDSL systems between customerpremises equipment (CPE), such as xDSL modems, and a CO, which mayinclude, for example, xDSL Access Multiplexer (DSLAM), xDSL line cards,and other equipment. For purposes of nomenclature used herein, the terms“far end users,” “users,” and “CPE” may be used interchangeably.Furthermore, as generally known by those skilled in the art, the termShannon channel capacity refers to the maximum amount of informationthat can be reliably transmitted over a given channel. Precoding is onetechnique used to mitigate the effects of crosstalk in an xDSL system.Essentially, interference caused by crosstalk is pre-compensated for (orcanceled out) at the CO before transmission of data.

Embodiments of off-diagonal multiple input/multiple output precoders(ODMP) are described herein which reduce self-FEXT effects from adjacentchannels in an xDSL system while ultimately increasing the Shannonchannel capacity. Furthermore, embodiments of ODMP described hereinreduce the effects of FEXT without consuming an excessive amount ofsystem resources. The various embodiments of ODMP described hereinachieve cooperative self-FEXT cancellation through various phases: (a)learning the xDSL MIMO channel and/or the MIMO precoder; (b) trackingsmall variations of the channel and/or precoder (for a fixed number ofusers); and (c) “incrementing” the ODMP to add new users (or removeexisting users) within the system. It should be appreciated that thesesteps are accomplished while meeting the following importantrequirements: 1) the overall objective of canceling or minimizing theeffects of self-FEXT; 2) minimal impact on the overall transmit power ofthe system; 3) use of linear-adaptive processing without the need forcomputationally intensive matrix inversion operations; 4) minimizationof the back channel bandwidth used for conveying error data; and 5) theneed for fast convergence toward the optimum ODMP while maintainingrobustness with respect to finite precision.

Reference is now made to FIG. 1A, which illustrates an xDSL system inwhich embodiments of ODMP are applied. As shown in FIG. 1A, the MIMOxDSL channel is denoted by the [N, N] matrix: H[q,t] at DMT-symbol timeinstant t and discrete frequency q. For the non-limiting DMT based xDSLsystem depicted in FIG. 1A, there are N separate users (or N sets of CPE110 a, 110 b, 110 c). The expression X[q,t] denotes the complex vectorcollecting the N QAM downstream transmitted raw symbols, prior topre-coding. Each of the QAM raw symbols is expressed as X_(m)[q,t],1≦m≦N at DMT symbol time instant t and discrete frequency q, fortransmitter m, 120 a, 120 b, 120 c. Finally, embodiments of MIMO linearprecoders described herein (i.e., ODMP) are expressed by the [N, N]matrix operator P[q,t] An ODMP module 132, which incorporates theoperator is generally implemented within the CO 130.

As described above, generally, the CO may include an xDSL Access

Multiplexer (DSLAM), ADSL line cards 140 a, 140 b, 140 c, and otherequipment for interfacing with CPE 110 a, 110 b, 110 c. The DMT basedxDSL system shown in FIG. 1A further includes a low speed back channel160 a, 160 b, 160 c used by each of the users (i.e., N sets of CPE 110a, 110 b, 110 c) to send feedback data upstream back to the CO 130. Thefeedback data sent from each user is used to derive an optimum ODMPP[q,t] and is denoted by the function θ_(m)[q,t], 1≦m≦N.

One perceived shortcoming associated with existing approaches is theneed to either perform matrix inversion operations on multiple matricesor the need to incorporate multiple linear systems, both of which arecomputationally intensive, regardless of whether processing is performedat the CPE side or the CO side. Furthermore, another perceivedshortcoming is the lack of an efficient means for “incrementing”precoders in order to add new users to the system. Embodiments describedherein address these perceived shortcomings by incorporating linearadaptive processing without the need for computationally intensivematrix inversion operations. Embodiments described herein also provide amessaging protocol that allows error data to be conveyed from thecustomer premises equipment (CPE) to the CO over a low speed backchannel. The messaging protocol is utilized to estimate and update theMIMO precoders at the CO.

Because the CPE individually perform demodulation of received signals,the diagonal terms within the MIMO xDSL channel matrix are separatelyand independently compensated, on a per tone basis, by a one tapfrequency domain equalizer FEQ performed on every CPE. Embodiments ofthe ODMP module 132 perform learning, tracking and incrementing (of newusers) to derive precoders that minimize the effects of self-FEXT.Exemplary embodiments of ODMP modules 132 further incorporate a reducedcost (least mean square) LMS-based algorithm that converges towards anoptimum ODMP function for maximizing the distributed Shannon capacity ofall the far end users (CPE).

The adaptive algorithm described herein utilizes error samples sent fromthe end users. As described herein, under correct detection and perfectdiagonal equalization assumptions, concurrently minimizing the end usererror variances results in maximization of Shannon capacities for eachuser. It should be noted that these assumptions are typically viable forxDSL systems since xDSL systems generally operate at a bit error rate of10⁻⁷ or better and use learning sequences that are thousands of symbolsin length. Furthermore, exemplary embodiments of ODMP utilize 16-bitprecision for the encoding of complex valued error data. It should beappreciated that such precision allows the adaptive scheme to convergeto a steady state solution in as little as 150 to 200 iterations whileachieving close to FEXT-free performance of 0.2 dB or less.

Finally, for embodiments described herein, in CO-based implementationsof cooperative FEXT cancellation, error data is passed from the CPE backto the CO for subsequent processing. A low speed back channel 160 a-c isutilized by the CPE 110 a, 110 b, 110 c to send error data back to theCO 130. For embodiments described herein for ODMP, a back channel withbandwidth on the order of only 128 kbps is sufficient to perform thelearning phase (associated with ODMP) for 1,000 tones on the order of afew seconds while the xDSL connection remains active. This enablesupdating of 1,000 ODMP every 125 ms.

Reference is now made to FIG. 1B, which illustrates various componentsof an alternative embodiment of the ODMP module 132 shown in FIG. 1A.For some embodiments, the ODMP module 132 located within the xDSL system130 in FIG. 1A and may be comprised of an initialization module 170, atracking module 172, and an incrementing module 174. The initializationmodule 170 is configured to learn characteristics of channels locatedbetween the CO and the plurality of users 110 a-c within the xDSL systemto derive an initial ODMP. The tracking module 172 is configured toiteratively converge towards an optimum ODMP function from the initialODMP function to reduce downstream self-FEXT by executing a least meanssquare (LMS) adaptive algorithm. In exemplary embodiments, the ODMPfunction may be represented as a zero diagonal matrix with onlyoff-diagonal terms. Furthermore, the tracking module maximizes thedistributed channel capacity for the plurality of users while keepingthe transmit power of the xDSL system 130 almost constant. Theincrementing module 174 is utilized to add a new user to the xDSL systemduring “showtime.” In other embodiments, the incrementing module 174 isutilized to remove existing users from the system 130.

Reference is now made to FIG. 2, which depicts a top-level diagram foran embodiment of a method for deriving an optimum ODMP for downstreamFEXT mitigation. Block 210 begins with the learning phase, wherecharacteristics associated with a plurality of N users within the xDSLsystem are “learned” to determine an initial off-diagonal multiple inputmultiple output (MIMO) precoder (ODMP) for a given tone frequency, whilethe end-to-end connection is already in showtime. Next in block 220, thetracking or converging phase takes place where an optimum ODMP isderived from the initial ODMP based on optimizing criteria in order tocancel downstream self-FEXT for the plurality of N users, wherein theODMP is represented as a zero diagonal matrix with only off-diagonalterms. For exemplary embodiments, the optimizing criteria comprisesmaximizing the channel capacity for all the users for the given tonefrequency such that maximizing the channel capacity minimizes theself-FEXT of all the users and minimizing the increase in transmit powerof the xDSL system to implement the ODMP. Finally, in block 230, theincrementing phase is performed in the event that a new user is added tothe system. Alternatively, embodiments of the ODMP described herein mayalso adaptively remove existing users from the system.

A low complexity and adaptive ODMP that maximizes the distributedShannon channel capacity per tone of all far end users is now describedin detail. The ODMP structure combined with the “diagonal dominance” ofthe MIMO xDSL channels described herein lead to an optimum ODMP thatmaximizes the Shannon channel capacity while having negligible impact onthe transmit power. One should also note that for exemplary embodimentsof the ODMP, the adaptive scheme does not require any matrix inversionoperations, which are computationally intensive in nature.

For purposes of nomenclature used herein, the following notations shallbe understood as follows: G[q,t] and G[q,t] respectively denote anN-sized column vector and a [N, N] matrix, both referring to a tone(denoted by q) at DMT time symbol (denoted by t). Furthermore, thesymbol ∥ ∥_(F) denotes the Frobenius norm of a matrix such that ∥G∥_(F)

√{square root over (TR[G^(H)G])}. Reference is briefly made back to FIG.1A, which illustrates various expressions used to derive an expressionfor MIMO precoding for cooperative downstream xDSL self FEXTcancellation for one particular discrete frequency tone q, q∈{q₁, . . ., q_(Q)} at a particular DMT symbol time instant t. FIG. 1A illustratesone particular tone for one particular DMT time symbol.

Embodiments of ODMP cancel the effects of self-FEXT by performing thefollowing. Before transmission through a MIMO xDSL downstream channelH[q,t] (shown in FIG. 1A) the complex vector X[q,t] configured tocollect N QAM raw symbols (again, denoted by the expression X_(m)[q,t],1≦m≦N in FIG. 1A) transmitted by N downstream xDSL transceiversundergoes MIMO linear precoding, represented as P[q,t]. The signalsreceived by the far end users may therefore be expressed by thefollowing equation:Y[q,t]=(HPX)[q,t]+V[q,t]  Equation 1In the equation above, Y[q,t] represents the signals received by theusers. Furthermore, V[q,t] denotes the vector, which reflects bothexternal (i.e., alien) crosstalk and background noise. As known by thoseskilled in the art, the vector V[q,t] is normal and circularlydistributed. Furthermore, V[q,t] is a zero mean vector, and although itmay be spatially correlated (in the presence of external crosstalk),very short xDSL crosstalk time coherence along with stationaryassumptions lead to essentially zero correlation across different tonesand across different DMT time symbols. As briefly discussed above, itshould be noted that although the precoder is a MIMO operator, theend-to-end system cannot be treated on a system level as a full MIMOsystem because the users (i.e., N sets of CPE as shown in FIG. 1A)independently perform demodulation of received signals Y_(m)[q,t],1≦q≦N. That is, the far end users do not operate in a cooperativefashion. Rather demodulation of received signals is performed on auser-by-user basis in an independent fashion. Thus, for exemplaryembodiments of ODMP, each far end user transmits feedback information(again, denoted by the function θ_(m)[q,t], 1≦m≦N) to the CO via a lowspeed channel with a bandwidth on the order of a few hundreds kbps. TheCO then processes this feedback data to perform learning, tracking, andincrementing functions to derive an optimum precoder. It should be notedthat while some embodiments involve CO-centric processing, otherembodiments may involve CPE-centric processing where processing of errordata is performed locally at the CPE rather than being sent upstream.

The MIMO channel matrix of the MIMO xDSL downstream channel H[q,t] maybe further expressed by the following equation:H=((H)_(d)(I+C))  Equation 2C denotes a zero diagonal matrix that represents self-FEXT channels.Thus, by definition, the zero diagonal matrix C has zero diagonalentries and the following holds true: C=(C)_(nd) where ( )_(nd) denotesthe operator that zeroes out the diagonal entries and where the offdiagonal terms remain unchanged. One should note that the linearprecoders described herein reflect off-diagonal components. The ODMPdefined by the following equation reflects this characteristic: P=(I−R).This equation represents the family of linear precoders which define theoff-diagonal typo of MIMO precoder since by definition, the matrix Ralso has zero diagonal entries and the following holds true: R=(R)_(nd).

As noted above, embodiments of the present invention seek to meetvarious performance constraints, including the requirement of derivingan ODMP that has minimal impact on the overall transmit power of thexDSL system. With the transmit power denoted as γ, further calculationslead to the following expression, which reflects the relative change inthe average transmit power from the N co-located xDSL transceivers withODMP implemented:

$\begin{matrix}{\frac{\delta\gamma}{\gamma}\hat{=}{\frac{{\gamma\left( {R = 0} \right)} - {\gamma(R)}}{\gamma\left( {R = 0} \right)} = {- \frac{{R}_{F}^{2}}{N}}}} & {{Equation}\mspace{14mu} 3}\end{matrix}$The operator ∥ ∥_(F) again designates the Frobenius norm of a matrix.The equation above leverages the fact that, by definition TR[R]=0. Thus,the relative change in average transmit power |δγ/γ| with an ODMPimplemented will generally be on the order of less than 10⁻², providedthat the off diagonal matrix R maximizes the Shannon capacity of all thefar end users. Those with ordinary skill in the art will appreciate thatsuch a value (<10⁻²) is considered negligible in xDSL systems. It shouldbe further noted that therefore, no additional transmit powerconstraints must be incorporated into the diagonal matrix R providedthat the distributed Shannon capacity for all the end users ismaximized. It should also be noted that all the co-located xDSLtransmitters emit the same downstream power, denoted byE[|X_(m)|²]=σ_(X) ², 1≦m≦N, which leads to the relationship,γ(R=O)=σ_(X) ².

Thus, the “precoded” MIMO-DSL channel (denoted by {tilde over (H)}) withODMP implemented can be expressed by the following equation:{tilde over (H)}

HP=[(H)_(d)(I−(CR)_(d))]+[(H)_(d)(C−R−(CR)_(nd))]  Equation 4The equation above illustrates that the equivalent MIMO channel issub-divided into two sub-MIMO channels. The first sub-MIMO channel is a“FEXT-free” diagonal matrix. The second sub-MIMO channel is off diagonalin nature and reflects self-induced FEXT.

Various embodiments described herein seek to maximize the downstreamdistributed Shannon channel capacity by mitigating the effects of FEXT.The Shannon downstream channel capacity may be expressed by theparameter {tilde over (Ω)}_(m). The downstream capacity experienced bythe far end user m, 1≦m≦N, in the presence of an ODMP is given by thefollowing equation:

$\begin{matrix}{{{\overset{\sim}{\Omega}}_{m} = {\log_{2}\left( {1 + \frac{{H_{m,m}}^{2}{{1 - ({CR})_{m,m}}}^{2}\sigma_{X}^{2}}{{{H_{m,m}}^{2}{{C_{m}^{T} - R_{m}^{T} - {{row}_{m}\left\{ ({CR})_{nd} \right\}}}}^{2}\sigma_{X}^{2}} + \sigma_{Vm}^{2}}} \right)}},\mspace{79mu}{1 \leq m \leq N}} & {{Equation}\mspace{14mu} 5}\end{matrix}$wherein the term C_(m) ^(T) (respectively R_(m) ^(T)) denotes the row m,1≦m≦N of matrix C (respectively matrix R) and the term σ_(Vm) ²designates the power from external and background noises experienced byreceiver m. As discussed earlier, since the far end users do not performjoint demodulation of received signals, the full MIMO Shannon capacitycannot be used as an optimization criterion for mitigating the effectsof self-induced FEXT since the end users to not cooperatively demodulatethe received signals. Rather, exemplary embodiments of ODMP mitigateself-induced FEXT by independently maximizing the channel capacity ofeach of the individual downstream channels.

An ODMP scheme that maximizes the channel capacity of all the far endusers (and that ultimately minimizes the effects of self-FEXT) is provento be defined by the following relationship:R ^(o) =C−(CR ^(o))_(nd)  Equation 6The matrix R denotes the concatenation of N row vectors R_(m) ^(T),1≦m≦N. Each of the N row vectors R_(m) ^(T), 1≦m≦N precodes the N−1sources that cause FEXT to be induced into receiver m. Since the endusers do not cooperate (i.e., do not perform joint modulation ofreceived signals), a search is performed through the row vectors R_(m)^(T), 1≦m≦N to find the source that maximizes the capacities {tilde over(Ω)}_(m), 1≦m≦N of all end users. Next, the optimum vectors concurrentlyzero the complex gradient of the capacities as reflected by thefollowing operation:∇_(R) _(m) {tilde over (Ω)}_(m)=0, 1≦m≦N  Equation 7Next, the complex gradient is applied to the Shannon capacity {tildeover (Ω)}_(m). One should note that the row vector row_(m){(CR)_(nd)}and the complex scalar (CR)_(m,m) do not depend upon the components ofR_(m) because of the off diagonal feature of both matrices C and R. TheN relationships for the optimum vectors are derived as:R _(m) ^(oT) =C _(m) ^(T)−row_(m){(CR ^(o))_(nd)}, 1≦m≦N  Equation 8Next, applying this equation into the Shannon capacity expressionresults in cancellation of the self-inducted FEXT components, therebyeliminating FEXT for the given capacity (denoted by {tilde over (Ω)}_(m)^(o)) of a precoded channel. Thus, the maximum capacity wherein theeffects of FEXT is eliminated is represented by the followingexpression:

$\begin{matrix}{{{\overset{\sim}{\Omega}}_{m}^{o}\hat{=}{\log_{2}\left( {1 + \frac{{H_{m,m}}^{2}{{1 - \left( {CR}^{o} \right)_{m,m}}}^{2}\sigma_{X}^{2}}{\sigma_{Vm}^{2}}} \right)}},{1 \leq m \leq N}} & {{Equation}\mspace{14mu} 9}\end{matrix}$Furthermore, if the Optimum ODMP R^(o), solution of equation 6 isreplaced by its “first order approximation” equal to the matrix C forpurposes of simplicity, then a performance loss is incurred. Theassociated change in SNR denoted by SNR_(loss,dB) ⁽¹⁾ is represented bythe following:SÑR _(m,dB) ^((1)loss)=10 log₁₀(1+SNR _(m)∥row_(m){(C)_(nd)⁽²⁾}∥²)  Equation 10The loss in SNR may thus be represented by the following:

$\begin{matrix}{{\overset{\sim}{SNR}}_{m,{dB}}^{{(1)}{loss}} \leq {10{\log_{10}\left( {1 + {{SNR}_{m}\frac{\rho^{4}}{N}}} \right)}}} & {{Equation}\mspace{14mu} 11}\end{matrix}$In equation 11, ρ designates an upper bound of the Frobenius norm of thematrix C. The parameter ρ is generally significantly smaller than unityfor a wide range of frequencies within the deployment reach of VDSLsystems. It should be noted that the relative change in transmit poweris negligible, as discussed earlier. The average transmit power withprecoding implemented is expressed as:

$\begin{matrix}{{\frac{{\delta\gamma}^{(1)}}{\gamma}} \leq \frac{\rho^{2}}{N}} & {{Equation}\mspace{14mu} 12}\end{matrix}$As a non-limiting example, consider an xDSL system with a 25 pair binderoperating at 3 MHz at 1,000 feet of coupling distance. A typical valuefor the parameter ρ 99% of the time is approximately 2×10⁻¹ for theseoperating parameters. Applying the equation above reveals that therelative transmit power change as a result of implementing variousembodiments of ODMP is indeed less than 10⁻² in 99% of deploymentscenarios. As such, embodiments of ODMP meet the objective of minimizingthe impact to the overall transmit power of the system with ODMPincorporated.

It should be appreciated that for various embodiments of the ODMPdescribed herein, ODMP not only maximizes the capacities of each of thechannels but also minimizes the variances in error associated with eachof the individual channels at the far end. This generally holds truegiven the very low bit error bit (BER) in which xDSL modems operate andthe high level of stability typical in xDSL lines. As described earlier,it should further be noted that various embodiments may provide for aCPE-CO shared processing scheme where computations are performed in adistributed manner. As a non-limiting example, error data may beprocessed at each of the individual CPE to generate ODMP parameters,which may then be passed upstream to the CO to perform precoding. Forsome embodiments, the CPE may transmit data to the CO over a lowbandwidth channel.

As part of the ODMP scheme, an adaptive algorithm is performed based onerror data passed from the CPE. Executing the adaptive algorithm allowsestimating, tracking, and incrementing to be performed in order toderive an optimum ODMP for a given xDSL system. As part of the adaptivealgorithm, an initialization process is first performed. The adaptivealgorithm is initialized with a maximum likelihood (ML) estimate of thefirst order ODMP expressed as R⁽¹⁾

C. The adaptive scheme follows a classical least mean square (LMS)stochastic gradient approach. Initialization is performed in normal datamode but without precoding enabled. This absence of precoding in theinitialization phase is reflected by R

O. A ML estimate of a first order ODMP vector can then be derived asfollows:

$\begin{matrix}{{{\hat{C}}_{- m}^{ML}\lbrack T\rbrack} = {{- \frac{1}{\sigma_{X}^{2}}}\frac{1}{T}{\sum\limits_{k = 1}^{T}{{X_{- m}^{*}\left\lbrack t_{k} \right\rbrack}{E_{m}\left\lbrack t_{k} \right\rbrack}}}}} & {{Equation}\mspace{14mu} 13}\end{matrix}$In equation 13, X_(−m) denotes a vector of size N−1 that collects theN−1 downstream QAM symbols transmitted from the N−1 channels that causeself-FEXT in channel m. This reflects the initial ODMP resulting fromthe learning phase. One should note that this initial ODMP reflects amere cross correlation between the error samples and the transmit QAMsymbols.

Next, the adaptive algorithm operates in data mode but this time withprecoding (ODMP) enabled. The transient phase of the adaptive schemecompletes the learning phase while the scheme can later be leveraged totrack and increment an optimum ODMP. While the adaptive processing isperformed, the error data is expressed as:

$\begin{matrix}{{{E_{m}\left\lbrack {t/t^{\prime}} \right\rbrack} = \frac{\left( {{\begin{pmatrix}{{R_{- m}^{T}\left\lbrack t^{\prime} \right\rbrack} - C_{- m}^{T} +} \\{{row}_{- m}\left\{ {({CR})_{nd}\left\lbrack t^{\prime} \right\rbrack} \right\}}\end{pmatrix}{X_{- m}\lbrack t\rbrack}} + {W_{m}\lbrack t\rbrack}} \right)}{\left( {1 - {({CR})_{m,m}\left\lbrack t^{\prime} \right\rbrack}} \right)}},\mspace{79mu}{1 \leq m \leq N}} & {{Equation}\mspace{14mu} 14}\end{matrix}$In the equation above, t represents the current time instant and t′represents the previous updating time instant. The derivation of theadaptive scheme is based on the calculation of the gradient of thesquare modulus of the error E_(m)[t/t′] given in the equation above,with respect to R_(−m) for all the receivers, m∈{1, . . . , N}. Thismethodology is based on classical instantaneous stochastic gradientapproaches. It should be noted that with the error equation above, thefact that both the row vector (row_(m){(CR)_(nd)}) and the complexscalar (CR)_(m,m) do not depend on the components of R_(m) (due to theoff-diagonal structure of the two matrices C and R) means thatcalculations are greatly simplified. Indeed, the equation above does notrequire any matrix inversion and is proportional to the instantaneouscorrelation between the error and the vector of FEXT disturbersexpressed as:∇_(R) _(−m) (|E _(m) [t/t′]| ²)∝X* _(−m) [t]E _(m) [t/t′],1≦m≦N  Equation 15The equation above leads to the following concurrent recursive updatesof the vectors {circumflex over (R)}_(−m) for all the receivers, 1≦m≦N:{circumflex over (R)} _(−m) [t _(n+1) ]={circumflex over (R)} _(−m) [t_(n) ]−μ[t _(n+1) ]X* _(−m) [t _(n+1) ]E _(m) [t _(n+1) /t _(n) ], n≧0,1≦m≦N  Equation 16As described above, the adaptive algorithm performs initialization basedon ML estimates of the first order ODMP expressed as:{circumflex over (R)} _(−m) [t ₀ ]=Ĉ _(−m) ^(ML) [T]={circumflex over(R)} _(−m) ⁽¹⁾ [t ₀], 1≦m≦N  Equation 17Finally, the N concurrent LMS algorithms described by the equationsabove converge toward an optimum ODMP solution which provides themaximum capacity for a precoded channel while eliminating FEXT.Furthermore, it should be appreciated that the adaptive algorithmdescribed above exhibits linear complexity in the sense that it requiresonly 2×(N−1) complex multiplication operations for each receiver m,1≦m≦N.

It should also be appreciated that embodiments of ODMP provide fornon-disruptive incrementing of ODMP such that new users may be added.Generally, it will appreciated by those skilled in the art that theaddition of new channels within an xDSL system is more challenging thanremoving existing channels. During the tracking phase, the size of theMIMO precoder does not change. That is, a fixed number of off diagonalcomponents are updated in order to follow slow variations of thechannel. When new users are added, however, the size of the MIMO-DSLchannel grows. Furthermore, the number of self-FEXT precoders increases.

As a non-limiting illustration, the following describes the process bywhich a single user is added to the system. It should be noted that thesame scheme may be extended to add multiple new users. For someembodiments, the addition of new users is performed while operating indata mode whereby the existing N users benefit from a [N, N] dimensionalODMP {circumflex over (R)}_(N). When a new user is to be added, thenetwork management informs the cooperative self-FEXT device at the COthat a new user, denoted by the index N+1, is activated at time t_(new).Thus, at this point in time, the ODMP is incremented to accommodate thenew user. Between time t_(new) and t_(init), the incremented ODMP isinitialized and then incremented based on ML estimation. Finally, attime t_(init), the LMS is activated for tracking purposes. Thus, theincrementing process can be summarized as a two-stepprocess—initialization and adaptation of the new user.

The following steps are performed after activation of the new userduring time t_(new)≦t≦t_(init). The [N+1, N+1] ODMP {circumflex over(R)}_(N+1) is first initialized and reflects the new user designated asN+1. After initialization, the adaptive algorithm is performed tocomplete the estimation of {circumflex over (R)}_(N+1) and in order toenter the tracking phase. During both the initialization and thecompletion of the learning phase of {circumflex over (R)}_(N+1), thepower of the new user N+1 may follow a ramping pattern that can beexpressed by:σ_(X,N+1) ² [t _(new)]≦σ_(X,N+1) ² [t]≦σ _(X,N+1) ² [t _(end) ], t_(new) ≦t≦t _(end)  Equation 18

It should be noted that the steady increase in the transmit powerassociated with new user N+1 minimizes the impact to the other existingusers even though the learning and precoding of the new user has notbeen performed yet. At the end of the full learning phase at t=t_(end),new user N+1 transmits the same power as the other CO-ports providingthe same service, σ_(X,N+1) ²[t_(end)]=σ_(X) ².

{circumflex over (R)}_(N+1) is initialized with a ML estimate of itsfirst order approximation (i.e., {circumflex over (R)}_(N+1)⁽¹⁾=Ĉ_(N+1)). Straightforward “accumulation property” of matrix C_(N+1)revealed in the equation below greatly facilitates the initialization,

$\begin{matrix}{C_{N + 1} = \begin{bmatrix}C_{N} & C_{N,{N + 1}} \\C_{{N + 1},N}^{T} & 0\end{bmatrix}} & {{Equation}\mspace{14mu} 19}\end{matrix}$Indeed, as displayed in the equation above, the sudden occurrence of anew user requires the estimation of only 2N new cross couplingcoefficients, C_(m,N+1)[q], 1≦m≦N (components of vector C_(N,N+1)) andC_(N+1,k)[q], 1≦k≦N (components of vector C_(N+1,N) ^(T)). The first setC_(m,N+1)[q], 1≦m≦N reflects the FEXT interference from the new useronto the existing users while the second set C_(N+1,k)[q], 1≦k≦Nexpresses the FEXT interference generated by the existing users onto thenew user. Since precoding has not been performed yet for the new userN+1, the ongoing precoder {circumflex over (R)}_(N)[t], t≧t_(new) of theN existing users does not change the actual cross FEXT channel stemmingfrom user N+1 onto the existing users. Therefore, the ML estimation ofthe coefficients C_(m,N+1)[q], 1≦m≦N may be expressed as:

$\begin{matrix}{{{{\hat{C}}_{m,{N + 1}}^{ML}\lbrack q\rbrack} = {- \frac{\sum\limits_{n = 1}^{T}{{E_{m}\left\lbrack {q,{t_{new} + \delta_{n}}} \right\rbrack}{X_{N + 1}^{*}\left\lbrack {q,{t_{new} + \delta_{n}}} \right\rbrack}}}{\sum\limits_{n = 1}^{T}{{X_{N + 1}}^{2}\left\lbrack {q,{t_{new} + \delta_{n}}} \right\rbrack}}}},\mspace{79mu}{1 \leq m \leq n}} & {{Equation}\mspace{14mu} 20}\end{matrix}$In the equation above, the term X_(N+1)[q,δ_(n)+t_(new)] denotes thedownstream transmit symbol from user N+1, for tone q and DMT time symbolδ_(n)+t_(new). Furthermore, the term δ_(n), 1≦n≦T denotes that T DMTtime symbols increment with respect to the activation time t_(new) ofnew user N+1. The estimation of the coefficients C_(N+1,k)[q], 1≦k≦Nrequires the compensation of the precoder {circumflex over(R)}_(N)[t_(new)+δ], δ≧0 such that:

$\begin{matrix}{{{\hat{C}}_{{N + 1},N}^{T,{ML}}\lbrack q\rbrack} = {\left( {{- \frac{1}{T}}{\sum\limits_{n = 1}^{T}{{E_{N + 1}\left\lbrack {q,{t_{new} + \delta_{n}}} \right\rbrack}{{\overset{\sim}{X}}_{- {({N + 1})}}^{H}\left\lbrack {q,{t_{new} + \delta_{n}}} \right\rbrack}}}} \right)\left( {I - {{\hat{R}}_{N}^{ref}\lbrack q\rbrack}} \right)^{- 1}}} & {{Equation}\mspace{14mu} 21}\end{matrix}$In the equation above, the term {tilde over (X)}_(−(N+1)) can becalculated as {tilde over (X)}_(−(N+1))=X_(−(N+1))/σ_(X) ² and is thenormalized vector of size N from which the contribution of user N+1 hasbeen removed. Furthermore, the term {circumflex over (R)}_(N) ^(ref)reflects any kind of “reference” for the matrices {circumflex over(R)}_(N)[t_(new)+δ_(n)], 1≦n≦T. This reference can be the original value{circumflex over (R)}_(N)[t_(new)], the final value {circumflex over(R)}_(N)[t_(new)+δ_(T)], an average, etc. Because of the smallvariations properties of the matrix {circumflex over (R)}_(N) ^(ref),the matrix inversion (I−{circumflex over (R)}_(N) ^(ref))⁻¹ that showsup in Equation 50 above can be efficiently replaced by the followingmatrix: ≈(I+{circumflex over (R)}_(N) ^(ref)). The initialization of{circumflex over (R)}_(N+1) may be expressed as:

$\begin{matrix}{{{{\hat{R}}_{N + 1}\left\lbrack t_{0} \right\rbrack} = \begin{bmatrix}{\hat{R}}_{N}^{ref} & {\hat{C}}_{N,{N + 1}}^{ML} \\{\hat{C}}_{{N + 1},N}^{T,{ML}} & 0\end{bmatrix}},{t_{0} \geq {t_{new} + \delta_{T}}}} & {{Equation}\mspace{14mu} 22}\end{matrix}$The subsequent updates to the row vectors of matrix {circumflex over(R)}_(N+1) are derived from Equation 16 above and yields the following:{circumflex over (R)} _(−m) [t _(n+1) ]={circumflex over (R)} _(−m) [t_(n) ]−μ[t _(n+1) ]X* _(−m) [t _(n+1) ]E _(m) [t _(n+1) /t _(n)], t_(n)≧t ₀ ≧t _(new)+δ_(T), 1≦m≦N+1  Equation 23It should be noted that in the equation above, both vectors R_(−m) andX_(−m) now have a size equal to N. Finally, while the steps describedabove relate to the addition of a single new user, multiple users may beadded as well.

Reference is now made to FIG. 3, which illustrates how a low speed backchannel may be used to incorporate an embodiment of the method shown inFIG. 2. Specifically, FIG. 3 provides simulation results reflecting thegain in downstream SNR for QAM tone #696 (3 MHz) at a loop length of 1kft over 26 AWG across varying levels of precision for the errorfeedback data. For the simulation results shown in FIG. 3, the transmitpower spectral density (PSD) is flat at −60 dBm/Hz, reflecting band plan998. The reference situation reflects data mode without precoding. Thereference raw SNR of tone #696 is equal to 43 dB. An ideal conditionrefers to a precoded channel that is FEXT-free. The ideal gain in SNR isderived from the difference (measured in dB) between the SNR under idealconditions and the SNR obtained when no precoder is used (shown to be 43dB in FIG. 3). The ideal SNR gain is equal to 23.4 dB.

FIG. 3 illustrates use of an ODMP I−{circumflex over (R)} where{circumflex over (R)} is based on Equations 16 and 17. The LMS algorithmis initialized with the ML value obtained (or converged upon) after Taverages or iterations. This convergence scheme is denoted byML(T)-LMS(K), where K stands for the number of iterations of the LMSupdating scheme. The SNR gain in the practical situation may beevaluated assuming regular precision or a smaller number of bits b tocode the real part and the imaginary parts of the error (i.e., (2×b)bits to code the complex error). For purposes of nomenclature, thepractical SNR gain is denoted by PG-SNR (b, T, K).

FIG. 3 displays the following four practical SNR gains versus the numberof LMS iterations K during the tracking phase, post learning phase forT=50 averages:

-   -   PG-SNR (b=6 bits, T=50, K)    -   PG-SNR (b=8 bits, T=50, K)    -   PG-SNR (b=16 bits, T=50, K)    -   PG-SNR (b=64 bits, T=50, K)

The learning phase is performed during “showtime” with no precoderactive while the tracking phase is performed with the precoder active.The four SNR figures shown in FIG. 3 reflect four different precisionlevels for the error, namely b=6 bits, b=8 bits, b=16 bits (all in fixedpoint format) and b=64 bits. The ideal SNR gain of 23.4 dB is denoted bythe horizontal line shown in FIG. 3.

Examination of FIG. 3 reveals that after performing the learning phasebased on only 50 points, it is possible to achieve a gain in SNR ofapproximately 14 dB while utilizing error precision levels of 64 bits,16 bits, and 8 bits. Furthermore, a gain in SNR of approximately 13.2 dBcan be achieved while utilizing an error precision level of 6 bits.Thus, it should be noted that utilizing just 8 bits of precision forerror feedback data yields the same SNR gain as when utilizing 16 bitsand even 64 bits of error precision.

After the learning phase, the FEXT cancellation based on embodiments ofthe ODMP precoder is enabled, and the tracking phase begins using theLMS scheme initialized by the cross channel “learned” values. It shouldbe further noted that as shown in FIG. 3, the LMS scheme utilized forembodiments of the ODMP described herein exhibits fast convergence forany precision level. As shown in FIG. 3, the LMS scheme convergestowards steady state in approximately 200 iterations for all four errorprecision levels (6 bits, 8 bits, 16 bits and 64 bits). Furthermore, useof error precision levels of 8 bits, 16 bits and 64 bits allows the LMSscheme to converge to as close as 0.2 dB within FEXT-free performance(i.e., ideal conditions). Thus, for exemplary embodiments of the ODMP,only 3,200 bits of error data are required to achieve adaptive ODMPconvergence. One skilled in the art will appreciated that 3,200 bits isjust a fraction of the number of bits required for other conventionalapproaches.

Regarding the convergence time of the learning and tracking phases forCO-based processing, it is assumed that the back channel operates on aper DMT symbol basis where error samples are transmitted in incrementsof 250 μsec such that each error sample contains 16 bits (8 bits foreach of the real and imaginary parts) and such that each DMT symbol cantransport the error samples of 2 tones. Based on these assumptions, atotal of only 6.25 ms is needed to perform 50 average samples under theML learning phase for one tone (i.e., 6.25 s for 1,000 tones). For thetracking phase, the convergence time needed to perform 200 LMSiterations for 1,000 tones is 25 seconds.

It should be noted that the learning phase is performed only once uponinitialization of the system. Once the system enters the tracking phase,the system does not need to re-enter the learning phase, even if usersare added or removed. Furthermore, it should be noted that the systemremains in “show time” with precoding enabled.

It should be emphasized that the above-described embodiments are merelyexamples of possible implementations. Many variations and modificationsmay be made to the above-described embodiments without departing fromthe principles of the present disclosure. Furthermore, it should beemphasized that embodiments discussed herein may be implemented in(and/or associated with) one or more different devices. Morespecifically, depending on the particular configuration, the embodimentsof the ODMP described herein may be implemented in any xDSL modem, COequipment such as an xDSL Access Multiplexer (DSLAM), xDSL line cards,and other equipment. All such modifications and variations are intendedto be included herein within the scope of this disclosure and protectedby the following claims.

1. A method for precoding data for transmission in a discrete multi-tone(DMT) system to cancel self-induced far end crosstalk (self-FEXT)comprising: learning, by the system, characteristics associated with aplurality of N users within a digital subscriber line (xDSL) system todetermine an initial off-diagonal multiple input multiple output (MIMO)precoder (ODMP) for a given tone frequency; and converging towards anODMP from the initial ODMP to cancel self-FEXT for the plurality of Nusers, wherein the ODMP is represented as a zero diagonal matrixcontaining only off-diagonal terms.
 2. The method of claim 1, whereinconverging towards an ODMP comprises maximizing a channel capacity forall the users such that variances in error data associated with theplurality of users is minimized.
 3. The method of claim 1, whereinlearning is performed during showtime and with the ODMP not enabled. 4.The method of claim 1, wherein converging is performed during showtimeand with the ODMP enabled.
 5. The method of claim 1, wherein the initialODMP is a cross correlation coefficient between data transmitted from aCentral Office (CO) and error associated with the transmitted data foreach user m, wherein the cross correlation coefficient is calculatedsuch that FEXT from all users other than user m is considered, wherein1≦m≦N.
 6. The method of claim 1, wherein converging towards the ODMPfurther comprises executing a least means square (LMS) adaptivealgorithm while the ODMP is enabled, the LMS adaptive algorithmexpressed by the following:{circumflex over (R)} _(−m) [t _(n+1) ]={circumflex over (R)} _(−m) [t_(n) ]−μ[t _(n+1) ]X* _(−m) [t _(n+1) ]E _(m) [t _(n+1) /t _(n)], n≧0,1≦m≦N wherein E_(m)[t_(n+1)/t_(n)] represents an error function, mrepresents a user, and X*_(−m) denotes a vector of size N−1 thatcollects N−1 downstream QAM symbols transmit from N−1 users that causeself FEXT into user m.
 7. The method of claim 6, wherein convergingtowards an ODMP further comprises receiving error data at a centraloffice (CO) from the users, wherein the error data is used by the CO toadaptively update the ODMP and converge towards the ODMP at the CO, andwherein calculating the ODMP is performed without matrix inversionoperations.
 8. The method of claim 7, wherein the CO receives the errordata from the users via a low speed back channel with a bandwidth ofapproximately 128 kbps.
 9. The method of claim 7, wherein the error datais received in a compressed form.
 10. The method of claim 6, whereinconverging towards an ODMP comprises processing the error data andperforming the least means square (LMS) adaptive algorithm for each ofthe N users.
 11. The method of claim 1, further comprising incrementingthe ODMP to add one or more new users to the xDSL system, whereinincrementing comprises estimating new coupling coefficients for each newuser comprising any one or both of a first set and a second set, whereinthe first set reflects FEXT from the new user into existing users, andwherein the second set reflects FEXT from other users into the new user.12. The method of claim 11, wherein the other users comprise onlyexisting users, only new users, or a combination of existing users andnew users.
 13. The method of claim 11, wherein estimating the newcoupling coefficients is performed according to the followingexpressions:$\mspace{79mu}{{{{\hat{C}}_{m,{N + 1}}^{ML}\lbrack q\rbrack} = {- \frac{\sum\limits_{n = 1}^{T}{{E_{m}\left\lbrack {q,{t_{new} + \delta_{n}}} \right\rbrack}{X_{N + 1}^{*}\left\lbrack {q,{t_{new} + \delta_{n}}} \right\rbrack}}}{\sum\limits_{n = 1}^{T}{{X_{N + 1}}^{2}\left\lbrack {q,{t_{new} + \delta_{n}}} \right\rbrack}}}},{1 \leq m \leq N}}$${{{\hat{C}}_{{N + 1},N}^{T,{ML}}\lbrack q\rbrack} = {\left( {{- \frac{1}{T}}{\sum\limits_{n = 1}^{T}{{E_{N + 1}\left\lbrack {q,{t_{new} + \delta_{n}}} \right\rbrack}{{\overset{\sim}{X}}_{- {({N + 1})}}^{H}\left\lbrack {q,{t_{new} + \delta_{n}}} \right\rbrack}}}} \right)\left( {I - {{\hat{R}}_{N}^{ref}\lbrack q\rbrack}} \right)^{- 1}}};$wherein C_(m,N+1)[q] represents FEXT generated by new user N+1 intoexisting user m; E_(m)[q,t_(new)+δ_(n)] represents an error function;X_(N+1)[q,t_(new)+δ_(n)] denotes a downstream transmit symbol from userN+1, for tone q and DMT time symbol δ_(n)+t_(new); δ_(n) represents anincrement of DMT time symbols with respect to an activation time t_(new)of new user N+1; and C_(N+1,N)[q] represents FEXT generated by N otherusers into user N+1 and {circumflex over (R)}_(N) ^(ref) represents areference.
 14. The method of claim 11, wherein incrementing furthercomprises limiting impact into existing users, wherein power for a newuser follows a steady increase profile optimized such that a change inSNR (signal-to-noise ratio) experienced by existing users is limited toa maximum value, wherein the maximum value is equal to a predeterminedmargin of the xDSL system.
 15. An off-diagonal multiple input multipleoutput (MIMO) precoder (ODMP) system for generating precoded signals toincrease system performance for a plurality of users in a digitalsubscriber line (xDSL) system, comprising: an initialization moduleconfigured to learn characteristics of channels located within the xDSLsystem associated with the plurality of users to derive an initial ODMP;and a tracking module configured to converge towards an ODMP from theinitial ODMP to reduce downstream self-induced far end crosstalk(self-FEXT) by executing a least means square (LMS) adaptive algorithm,wherein the ODMP is represented as a zero diagonal matrix containingonly off-diagonal terms.
 16. The system of claim 15, wherein thetracking module is further configured to maximize the channel capacityfor the plurality of users by minimizing variances in error dataassociated with each of the plurality of users.
 17. The system of claim15, wherein the initialization module learns characteristics of channelsduring showtime and with the ODMP not enabled.
 18. The system of claim15, wherein the tracking module is further configured to receive errordata from each of the plurality of users, wherein the error data is usedto calculate an updated ODMP, and wherein calculating the updated ODMPis performed without matrix inversion operations.
 19. The system ofclaim 18, wherein the tracking module receives error data from theplurality of users via a low speed back channel with a bandwidth ofapproximately 128 kbps.
 20. The system of claim 18, wherein the errordata received in a compressed form.
 21. The system of claim 15, furthercomprising an incrementing module configured to add one or more newusers to the xDSL system during showtime.
 22. An off-diagonal MIMOprecoder (ODMP) in a system for pre-compensating for downstreamself-induced crosstalk (self-FEXT) comprising: a learning moduleconfigured to learn CPE channel characteristics to obtain an initialODMP; a tracking module configured to update the initial ODMP duringshowtime to obtain an ODMP by receiving error data over a feedbackchannel and executing a least means square (LMS) adaptive algorithm; andan incrementing module configured to perform one of: add one or more newCPE to the system and remove existing CPE from a digital subscriber line(xDSL) system.